2004 Fiscal Year Final Research Report Summary
Algebraic properties of homotopy classes in homotopy theory
| Project/Area Number |
14540063
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Chiba University |
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Principal Investigator |
MARUYAMA Kenichi Chiba University, Faculty of Education, Associate Professor, 教育学部, 助教授 (70173961)
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| Co-Investigator(Kenkyū-buntansha) |
KOSHIKAWA Hiroaki Chiba University, Faculty of Education, Professor, 教育学部, 教授 (60000866)
YAMAUCHI Kenichi Chiba University, Faculty of Education, Professor, 教育学部, 教授 (20009690)
TSUKIYAMA Kouzou Shimane University, Faculty of Education, Professor, 教育学部, 教授 (20093651)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Algebraic topology / Homotopy theory / Homotopy sets / Automorphism groups / Nilpotent groups |
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| Research Abstract |
Homotopy classes of maps of ten have binary operations. It is very useful to study their algebraic properties to know the topology of spaces. Homotopy groups and (co) homology groups are typical examples of such algebraic structures.
In this project we dealt with the self-homotopy sets of spaces. In this case homotopy sets are monoid by the binary operation induced by composition of maps. It is well known that studying composition of maps is very important in homotopy theory. Here mainly we study two kinds of subsets of self-homotopy sets. First we consider natural subgroups of self-homotopy sets which consists of maps inducing the trivial map on homotopy. Secondly we study the subset of self-homotopy equivalences. The subset inducing the trivial map on homotopy is known to be a nilpotent semigroup. We have determined the nilpotency of these semigroups in the case where spaces are rank 2 Lie groups and simply connected Hopf spaces. Moreover these semigroups define a filtration on a self
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-homotopy set. It is known that the filtration has finite length. We defined numerical invariants sz(X) and lz(X) for a space X. Then we have determined these two numbers for compact Lie groups. Next we summarize our results on self-homotpy equivalences. For any space the homotopy classes of self-homotopy equivalences is a group. It is neither abelian nor nilpotent in general. However subgroups associated with homotopy groups which are defined similarly as for the above case are known to be nilpotent groups. Nilpotent groups have nice properties like abelian groups. In particular there exists localization theory for nilpotent groups. We can transfer some results obtained for the sets which consists of maps inducing the trivial map on homotopy to self-homotopy equivalences group by using localization theory.
Through this study we have obtained the results above. Further, we have realized some new important problems in this project which we will study in future.
The investigators have contributed to the project by their considerations from their special fields. Less
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